$T: V \to V$ is a linear transformation such that $\operatorname{Range}(T)=\operatorname{Nullspace}(T)$. Which of the following is correct?

Question

Suppose $V$ is a finite dimensional non-zero vector space $\mathbb C$ and $T: V \to V$ is a linear transformation such that $\operatorname{Range}(T)=\operatorname{Nullspace}(T)$. Then which of the following is true?

1. The dimension of $V$ is even.
2. $0$ is the only eigenvalue of $T$
3. Both $0$ and $1$ are the eigenvalues of $T$.
4. $T^2=0$

I really have no idea how to approach this problem. Can someone please help me figure out how to solve the problem?

Answer 1

$(a)$ follows from Rank-Nullity theorem.
Next, try to show $T^2=0$ from the given condition, which will show $(b)$ is true but $(c)$ is not.